It is well known that the core of a game may be empty. The implication of this fact
for the Coase theorem is discussed in two papers, Aivazian & Callen (1981) and Coase
(1981).

Aivazian and Callen give an example with three firms, where the profits of the possible of firms are such that the core is empty so for all feasible distributions of profits there will exist a coalition (consisting of one, two, or three firms) who can get a higher profit.

From this they conclude »that the Coase theorem cannot be proved if the core of the
game is empty« (pp. 175-176).

In his »Comment« Coase states that the example by Aivazian and Callen »has not
led me to modify my views« (p. 183).

The part of the Coase theorem under discussion is the trivial observation made already Wicksell in the context of competitive equilibrium in his review of Pareto's Manuel d'Economie Politique, that if the coalition of all firms (players, agents, consumers without transactions cost can make any joint decision on the distribution of profits (on strategies of a game, on reallocation of initial resources, etc.), then any equilibrium be optimal.

It is an error in logic, that Aivazian and Callen from their example conclude, that the Coase theorem is incorrect. What their example shows is that the set of equilibria may be empty. The Coase theorem for the cases, where the set of equilibria is empty, is trivially simply because the elements of the empty set have all properties, they are also optimal.

The »Comment« shows in my opinion, that the theoretical world in which Coase is
living, is a world without existence of equilibria. The set of actions available to the

Resumé

SUMMARY: An example given by Aivazian and Callen of an empty core does not show
that the Coase theorem is incorrect, it shows that the theorem may be empty in the sense
its conclusion is about elements in an empty set.

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This paper was written March 1983, but never published. The 1991 Nobel prize to Coase may give some interest to this application of a little logic to the ideas of Coase.

firms and the coalitions of firms in his discussion of the example is so large that no
equilibria can exist.

Only under very special assumptions will an economy, a game, or a social system, where all the agents and all the subsets of agents - coalitions - can choose any preferred have an equilibrium. A precise formulation and precise results can be found in Vind (1983) - especially Theorems 1 and 2 - and Keiding (1985).

If one wants an economy, a game, or a social system for which both the Coase theorem and equilibria exist one should, except for a few special cases, have to prevent and »coalitions« of two from doing what is best for them, unless it is improving for everybody influenced by the action. Such assumptions are not realistic as a description of economic reality, and would be very difficult to enforce through a legal system.

References

Aivazian, VA. & J.L. Callen. 1981. »The Coase Theorem and the Empty Core«. Journal of Law and Economics 24, pp. 175-81.

Coase, R.H. 1981. »The Coase Theorem and
the Empty Core: A Comment«. Journal of
Law and Economics 24, pp. 183-187.

Keiding, H. 1985. »On the Existence of Equilibrium in Social Systems with Coordination«. Journal of Mathematical Economics 14, pp. 105-111.

Vind, K. 1983. »Equilibrium with Coordination«.
of Mathematical Economics